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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Averages

There are four types of average which need to be known: the mean, the median, the mode and the range.



The mean is what most people mean when they say 'average'. It is found by adding up the values of all of the pieces of data in a set and dividing by the total number of pieces of data.


Find the mean of 3, 5, 7, 3 and 5.

Mean = \dfrac{3+5+7+3+5}{5} = 23/5 = 4.6.

The mean for grouped data

When you are given data which has been grouped, the mean is \dfrac{\sum fx}{\sum f} , where f is the frequency and x is the midpoint of the group.


Work out an estimate for the mean height.

Height (cm) Number of People (f) Midpoint (x) fx
101-120 1 110.5 110.5
121-130 3 125.5 376.5
131-140 5 135.5 677.5
141-150 7 145.5 1018.5
151-160 4 155.5 622
161-170 2 165.5 331
171-190 1 180.5 180.5

\sum fx = 3316.5, \; \sum f = 23.

Mean = 3316.5/23 = 144cm (3s.f.).


The mode is the number in the set of data which occurs the most often. We sometimes call the mode the modal value.


Find the modal value of 5, 6, 3, 4, 5, 2, 5 and 3.

The modal value is 5, because there are more 5s than any other number.

NB: If there are two modes, the set of data is often called bimodal, or is said to have no mode.


The range is the largest number in a set minus the smallest number. It gives a measure of how spread out the data is.


Find the range of 5, 7, 9 and 14.

The range is 14 - 5 = 9.


The median of a group of data is the number in the middle, when the numbers are written in orer of size.


Find the median of 4, 1, 6, 2, 6, 7, 8.

First write the set of numbers in size order: 1, 2, 4, 6, 6, 7, 8.

The middle number is one of the 6s, so the median is 6.

Finding the middle number

If you have n numbers in a group, the median is the (n + 1)/2 th value.

For example, there are 7 numbers in the example above, so replace n by 7

and the median is the (7 + 1)/2 th value 4th value. The 4th value is 6.

If the set of data has an even number of pieces, then find the middle two pieces of data an work out the mean of these two numbers.


Find the median of 2, 4, 1, 10, 6, 8, 2 and 7.

First write the number is size order: 1, 2, 2, 4, 6, 7, 8, 10.

There are 8 pieces of data, so we find the middle 2 (the 4th and 5th pieces): 4 and 6.

Finally, find the mean of 4 and 6: (4+6)/2 = 10/2 = 5.

The median is 5.

Exam Tips

  • Averages do not have to equal one of the pieces of data. For example the median above was 5, but 5 wasn't one of the pieces of data.
  • Some sets of data have no mode - this occurs when two or more numbers occurs equally often. If asked to find the mode of some data like this, say there is no mode, but also say which the values are which occur most often.
  • You can think up many ways to remember which each of the different averages are. For example: median = middle number, mode = most common number.


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